Hankel matrices and lattice paths (Q2736384)

Hankel matrices \(H\) formed from a sequence of real numbers \(S= \{a_0=1,a_1,a_2,a_3,\dots\}\) and the lower triangular matrices \(L\) obtained from the Gaussian column reduction of \(H\) are considered. It is shown that the associated Stieltjes matrix \(S_L\) is a tridiagonal matrix. For any sequence of nonzero real numbers \(T= \{d_0= 1,d_1,d_2,d_3,\dots\}\) there are infinitely many sequences such that the determinant sequence of the Hankel matrix formed from these sequences is \(T\). The connection between the decomposition of a Hankel matrix and Stieltjes matrices and the connection between certain lattice paths and Hankel matrices are discussed. An explicit formula for the decomposition of a Hankel matrix is presented. The Gaussian column reduction for \(H\) is given.